Properties

Label 4-7600e2-1.1-c1e2-0-11
Degree $4$
Conductor $57760000$
Sign $1$
Analytic cond. $3682.82$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·7-s − 9-s − 2·13-s + 2·17-s + 2·19-s − 4·21-s − 2·23-s + 6·27-s + 2·29-s − 4·31-s + 16·37-s + 4·39-s + 4·43-s − 16·47-s − 9·49-s − 4·51-s + 2·53-s − 4·57-s − 26·59-s + 4·61-s − 2·63-s − 2·67-s + 4·69-s + 8·71-s + 22·73-s − 4·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s − 1/3·9-s − 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.872·21-s − 0.417·23-s + 1.15·27-s + 0.371·29-s − 0.718·31-s + 2.63·37-s + 0.640·39-s + 0.609·43-s − 2.33·47-s − 9/7·49-s − 0.560·51-s + 0.274·53-s − 0.529·57-s − 3.38·59-s + 0.512·61-s − 0.251·63-s − 0.244·67-s + 0.481·69-s + 0.949·71-s + 2.57·73-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57760000\)    =    \(2^{8} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3682.82\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 57760000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 16 T + 130 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 26 T + 285 T^{2} + 26 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 124 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 22 T + 259 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.67231863420945891258511979446, −7.61771084735817423262311289446, −6.85845465646839794967814182019, −6.51267623496176993291326644980, −6.22547758500510876648110098454, −6.11505944747201651727072693718, −5.46872027428920782492720767828, −5.26710488766648331407377673434, −4.99881897894251650911263695977, −4.70522009819694583292009766514, −4.16711607889211427801941163521, −3.90579103523269888896598860152, −3.20673599428109798944918002263, −2.94072556248092215428615500613, −2.49838564805811716779720436384, −1.98525114463207343413458412028, −1.25348210809507190295900154157, −1.13896275890434627643294934888, 0, 0, 1.13896275890434627643294934888, 1.25348210809507190295900154157, 1.98525114463207343413458412028, 2.49838564805811716779720436384, 2.94072556248092215428615500613, 3.20673599428109798944918002263, 3.90579103523269888896598860152, 4.16711607889211427801941163521, 4.70522009819694583292009766514, 4.99881897894251650911263695977, 5.26710488766648331407377673434, 5.46872027428920782492720767828, 6.11505944747201651727072693718, 6.22547758500510876648110098454, 6.51267623496176993291326644980, 6.85845465646839794967814182019, 7.61771084735817423262311289446, 7.67231863420945891258511979446

Graph of the $Z$-function along the critical line